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Basis beeldverwerking (8D040) d r. Andrea Fuster Prof.dr . Bart ter Haar Romeny dr. Anna Vilanova . Marcel Breeuwer. Filtering. Contents. Sharpening Spatial Filters 1 st order derivatives 2 nd order derivatives Laplacian Gaussian derivatives

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Basis beeldverwerking (8D040)

dr. Andrea FusterProf.dr. Bart terHaarRomenydr. Anna Marcel Breeuwer


  • Sharpening Spatial Filters
    • 1st order derivatives
    • 2nd order derivatives
    • Laplacian
    • Gaussian derivatives
    • Laplacian of Gaussian (LoG)
    • Unsharp masking
sharpening spatial filters
Sharpening spatial filters

Image derivatives (1st and 2nd order)

Define derivatives in terms of differences for the discrete domain

How to define such differences?

1 st order derivatives
1st order derivatives
  • Some requirements (1st order):
    • Zero in areas of constant intensity
    • Nonzero at beginningof intensity step or ramp
    • Nonzero along ramps
2 nd order derivatives
2nd order derivatives
  • Requirements (2nd order)
    • Zero in constant areas
    • Nonzero at beginningand end of intensity step or ramp
    • Zero along ramps of constant slope
image derivatives
Image Derivatives






1st order

2nd order


1st order

2nd order

Zero crossing, locating edges


Edges are ramp transitions in intensity

    • 1st order derivative gives thick edges
    • 2nd order derivative gives double thin edge with zeros in between
  • 2nd order derivatives enhance fine detail much better

1st order

2nd order

Zero crossing, locating edges

filters related to first derivatives
Filters related to first derivatives

Recall: Prewitt filter, Sober filter (lecture 2 – 01/05)

laplacian second derivative
Laplacian – second derivative

Enhances edges



Opposite sign for second order derivative

Adding diagonal derivation


Note: Laplacian filtering results in + and – pixel values

Scale for image display

So: take absolute value or positive values

line detector
Line Detector


Positive values Laplacian

(figure 10.5 book)


image sharpening example
Image sharpening - example

C=+1 or -1

Enhanced + Laplacian x8

8-connected Laplacian

Enhanced + Laplacian x5

Enhanced + Laplacian x6

4-connected Laplacian

Better sharpening with 8-connected Laplacian

(see figure 3.38 (d)-(e) book)

filtering in frequency domain
Filtering in frequency domain
  • Basic steps:
      • image f(x,y)
      • Fourier transform F(u,v)
      • filter H(u,v)
      • H(u,v)F(u,v)
      • inverse Fourier transform
      • filtered image g(x,y)
laplacian in the fourier domain
Laplacian in the Fourier domain


Fourier domain

blur first take derivative later
Blur first, take derivative later
  • Smoothing is a good idea to avoid enhancement of noise. Common smoothing kernel is a Gaussian.

Scale of blurring

gaussian derivative
Gaussian Derivative
  • Taking the derivative after blurring gives image g
gaussian derivative1
Gaussian Derivative
  • We can build a single kernel for both convolutions

Use the associative property of the convolution

laplacian of gaussian log
Laplacian of Gaussian (LoG)

LoG a.k.a. Mexican Hat

sharpening with log
Sharpening with LoG

sharpening with Laplacian

sharpening with LoG

unsharp masking highboost filtering
Unsharp Masking / Highboost Filtering
  • Subtraction of unsharp (smoothed) version of image from the original image.
    • Blur the original image
    • Subtract the blurred image from the original(results in image called mask)
    • Add the mask to the original

Let denote the blurred image

Obtain the mask

Add weighted portion of mask to original image


(see also figure 3.40 book)



unsharp mask

u.m. result

h.f. result

  • If
    • Unsharp masking
  • If
    • Highboost filtering
unsharp masking
Unsharp masking

Simple and often used sharpening method

Poor result in the presence of noise – LoG performs better in this case